Dynamic coupling of a liquid-tank system under transient excitations

Ocean Engng, Vol. 17, No 3, pp. 263-277, 1990. Printed in Great Britain.

0029-8018/90 $3.00 + .00 Pergamon Press plc

DYNAMIC COUPLING OF A LIQUID-TANK UNDER TRANSIENT EXCITATIONS

SYSTEM

ANDREW P u I - C H U N LUt FMC Corporation, San Jose, California, U.S.A.

and JACK Y. K. Lou Ocean Engineering Program, Texas A&M University, College Station, Texas, U.S.A. Abstract--It has been observed that the unrestrained free surface of a container can create relatively large liquid movements for even very small motions of the container. This excessive movement may endanger the stability as well as the maneuvering quality of the transporting vehicle. Therefore, the effects of the dynamic coupling of a liquid-tank system are of great concern. This dynamic coupling problem is studied analytically for a two-dimensional, rectangular rigid tank with no baffles. The governing equations of the liquid motion are derived with reference to a moving coordinate system which is fixed to the moving container. With the liquid forces generated by the fluid motion as the external exciting force for the tank, the motions of the liquid-tank system can be described according to Newton's Second Law of Motion. By using the Laplace transformation technique, the dynamic responses of the coupled system can be examined in detail. Numerical results for various types of external excitations and the resultant motions of the fluid-tank system are presented and compared with the equivalent non-shifting cargo system. The results of the comparison indicate that the discrepancy of responses in the two systems can obviously be observed when the ratio of the natural frequency of the fluid and the natural frequency of the tank is close to unity. Also, the amount of fluid inside the tank is a very important factor in determining the responses. INTRODUCTION

LIQUID sloshing in a moving container constitutes a broad class of engineering problems of great practical importance with regard to the safety of the transportation system. The unrestrained free surface of the liquid has an alarming propensity to undergo relatively large excursions for even very small motions of the container. This is particularly true for tank trucks on highways, tank cars on railroads, and sloshing of liquid cargo in ocean-going vessels. The chaotic liquid motion in a moving container will result in high impact pressures on the container structures and may create sufficient movement to overturn the vehicle. The problem of sloshing has been studied recently by many investigators. Earlier investigations are concerned with various types of oscillations of fuel tanks for spacecraft and roll stabilization tanks for ships. An extensive review of work accomplished prior to the mid-sixties is given by Abramson (1966). More recent results were summarized by Cox et al. (1979), Lou et al. (1980), and Hamlin et al. (1986). However, due to the lack of fundamental understanding of the effects of various parameters in the sloshing phenomenon, many problems remain unsolved. Chu et al. (1968) carried out an analytical and experimental study on ship roll stabilization tanks in which a mathematical model has been developed for rectangular 263

264

A. PuI-CHuN LuI and J. Y. K. Lot:

tanks executing a roll oscillation. Since linear analysis is employed, the results are valid for small amplitude fluid motion only. Visineau (1979) studied, experimentally, the dynamic effects of the entrapment of water on the deck of a ship in regular beam seas. It was observed that the presence of limited entrapped water on deck may damp out the rolling motion of the ship. However, if a large enough volume of water accumulates and the ship has only a small initial intact stability, the ship may capsize as a result of loss of static stability. Recently, Culley et al. (1978) completed a series of full-scale tests to examine how cargo shifting can affect a vehicle's handling quality. The tests were conducted by using vehicles with different loading conditions. The results of the tests show that the shifting cargo does affect the vehicle performance, in particular braking and maneuvering. A literature review reveals that although the problem of liquid sloshing in a rigid container has been studied by many investigators, few analytical studies have been conducted on the dynamic liquid-vehicle coupling effects. Therefore, the primary purpose of this study is to analytically investigate the effects of fluid sloshing on vehicle motions. GOVERNING EQUATIONS FOR THE FLUID A two-dimensional, rigid rectangular tank without baffles is partially filled with liquid and is forced to oscillate in a vertical plane containing the longitudinal center plane of the tank. The fluid is assumed to be incompressible and inviscid. Also, the energy losses due to the wave disintegration at the free surface are neglected. The moving coordinate system adopted by Lou et al. (1980) is used. This coordinate system is fixed to the oscillating tank with one of its axes coinciding with the undisturbed free surface as shown in Fig. 1. The motion of the fluid in the tank is governed by the continuity equation + w,,, = 0

(1)

and the equations of motion

J~

a ~

~

a ~1

I

I

• .......

~__~_.,

'-\

L..,,.y\\

....

EOUILIBRIUM POSITION

--,

INSTANTANEOUS POSITION

___L

Fz6. 1. Moving coordinate system.

/

J

Dynamic coupling of a liquid-tank system DVDt 2t)w = - g s i n 0 - ~P,y + z(~ + y02 _ f~c

265 (2)

D w + 2t)v = - g c o s 0 - -1P,z - y0 + z0'2 - zc Dt p where D o 0 0 Dt = Ot + Vffyy + Woz

(3)

and 0 is the angular displacement measured from the equilibrium position; 6 is the velocity and 0 is the acceleration of the rotational motion; p is the density and P is the pressure of the fluid; g is the gravitational acceleration; ~<.and ~c are the accelerations of the origin of the moving coordinate in the y and z directions, respectively. One of the advantages of using the moving coordinates system is that the boundary conditions at the fluid-tank interface are always homogeneous regardless of the magnitude of the tank motion, and are given by w= 0 v =0

atz = -D aty = -a

(4)

where D is the water depth and 2a is the length of the tank. At the free surface, the pressure must be zero. This leads to the dynamic free surface condition VP x ~ = 0

at z ='q

(5)

where -q = -q(y,t) is the surface elevation, and h is the unit vector normal to the free surface -q. The kinematic free surface condition states that "q,t + V'l],y = W

at z = -q.

(6)

Equations (1)-(6) completely define the sloshing problem under consideration. By introducing the stream function, +, such that v = - ~ , ~ and w = i~l,y

(7)

and eliminating the pressure P from Equation (2), the following equation of motion in terms of the stream function + can be obtained (V21~J),t- I~/,z(V21~l),y + I~l,y(V2~j),z = --26"

(8)

while the substitution of Equations (2) and (7) into Equations (5) and (6) yields rl,y [ - g c o s 0 - y0 + z02 - 2;c + 20+,z - ill,ty + IlJ,z(llJ,yy ) +,y(+,zz)] + [ - g s i n 0 + z0' + y02 - y,. + 26~,y + +,,~ - ~,~(~,~y) + ~,y(~,z~)] = 0 at z = rl -

(9)

and Tl,, -- *,z "q,y = O,y

at z = "q.

(10)

266

A. PuI-CttuN Lul and J. Y. K. Lou

Also, the boundary conditions of Equation (4) can now be written as 0,,, = 0 +,z = 0

atz=D aty = -+a.

(ll)

Since no assumptions have been made in deriving these equations regarding the smallness of the excitations, this set of equations is applicable to nonlinear liquid sloshing due to large amplitude excitations. However, for small amplitude motions, these equations may be linearized. Introducing the Laplace transform of a function, f(t), defined by Thomson (1950) f~(s) =

f7

e-~tf(t) dt = L~(t)}

(12)

the following linearized governing equations in the transformed plane are obtained: V2~ = -2L{O}

(13)

,: = 0 q~,y=

0

a t y = -+ a

(14)

at z = -D

(15)

s2(b ,z - g~ ,yy= gL{O} + L(Yc} at z = 0.

(16)

It can be shown that the solution for the stream function t~ in the transformed plane can be written as 2 ( - 1 ) ~ ~t

t} = -L{~}}Y2 +

L k = odd

cos ~xy

°/-3a

{

[2gc~L{0 } + ~ cosh ~D (L{Yc} - gL{~}})] sinh o~z (s 2 cosh a D + ga sinh aD) [-2s2L{0} + a sinh a D (L{Yc} - gL{0})] cosh az } (s~ cosh a D ~--~ ~nh ~ D )

+

(17)

where a = k~/2a. Similarly, the equation for the surface elevation, "q, can be obtained from the linearized kinematic free surface condition, s-~ - 0 ( 0 ) + : ~; ,,..

(18)

Assuming an initial condition "q(0) + in the form of "q(0) + = - y tan0o

(19)

where 0o is the initial angular displacement, the following solution for fi can readily be obtained (20)

1 (2L{0}y + y tan0o) 2(-1)~ -

sin ~y

k =odd

°¢2a

-2s2L{O} +~_ sinh c ~ D ( L ~ } _ - _ gL{~)})] s(s z cosh a D + got sinh otD) J"

I

267

Dynamic coupling of a liquid--tank system

HYDRODYNAMIC FORCES DUE TO THE FLUID MOTION The hydrodynamic pressure generated by the fluid motion can be evaluated from the equation P =

P,z dz

(21)

where P,z is obtained from the second equation of Equation (2). Thus, with the aid of Equation (17) the linearized expression for the pressure can be written in the transformed plane ~c

15 =29s L{O}yz + sp ~

sinay [Ak (coshotz-1) + B, sinhotz]

k = odd

\

/

(22) where Ak, Be are the coefficients for the cos ay sinh az and cos ay cosh az terms in Equation (17), respectively. The horizontal pressure force, FH, on the tank side wall can be obtained by integrating the pressures, and can be expressed as ~c

[:. = +--pbaD2s L{0} - spb ~

(-1) &vzi

k=odd

[Zk ( l s i n h a D - D ) + B k

+ pgb

(1

lcoshaD)]

(23)

+ ~D +- ~ pabD L{t}'}+ ~ pbD z L{~c}

where b is the width of the tank, the upper sign is to be used for y = a and the lower sign is for y = - a . Similarly, the vertical pressure force on the tank bottom is given by

F,, = 2pgDab 1_+ 2pDab L{2~}.

(24)

s

By going through similar procedures, the total moments M/t, created by the horizontal force and My, created by the vertical force, can be determined. They are given in the following equations:

~,1, = - 4~ 9D3abs L{0} + 23 paD3b L{tJ} - 2s pgD2ab L{~}} zc

1 pgD2a tan0o -

s

~] 29gDab 1-2sZ L{0} + otsinhotD ( L ~ } - gL{t)})] k

odd

( - 1 ) ~-t I A, (

- 29bs k =odd

J

a12

DsinhotD+ a ~1 - coshetD + ~ )

(25)

268

A. PuI-CHUN kul and J. Y. K. Lou

Mv= 34 pDba-s3

2

4

L{0} - 3 pDa3b L{{J} + 38 pga-~b L{0}

2

+ 3s 9ga3b tan0o + ~

49gb

(26)

k=odd o~4a

--2S L{0} + cxsinhcxD (L~,.} - gL{{~})] s(s:cosh~x-[) + gas~h-aD) J

I

~

2pbs (_l)g_~± [Ak (coshe~D - 1) - Bk sinh~xD].

k=o d d

0£2a

DYNAMIC COUPLING OF THE FLUID-TANK SYSTEM With the fluid pressure forces acting as the external exciting forces on the tank, the equations of motion for the tank can be derived by applying Newton's Second Law of Motion. Several types of motions are studied.

Case l--rectilinear motion of a fluid-tank system Consider a tank attached to an elastic spring and a dashpot. The tank is partially filled with a liquid and is free to oscillate on a smooth horizontal plane. We first investigate the oscillatory type of motion due to an initial displacement of the system. The subsequent tank motion and the effect of liquid sloshing are determined from the equation of motion for the tank Yc ( ms2

+ CS + K) - Yo (ms + c) =/enet

(27)

in the transformed plane, where Yo is the initial displacement and ~'net is the net horizontal force exerted on the tank wall by the fluid. Since only the translational motion is considered, 0 = 6 = 0 = 2c = 0 and the net horizontal force can be determined from Equation (23). The resultant expression can be written as

P,~et = L~,.} k ~ooo t ~'2as (s2 + Oa2)j

(28)

where

f,(k) = 4pb (lo tanhaD - D) f2(k) = 49goLDb tanhaD oa2 = ga tanhtxD.

(29)

Assuming that the initial velocity and acceleration at t = 0 are zero, the solution of ~, can be written as ~

.9, (ms 2 + cs + K) - k=oda ~

-- s2f2(k)] a2a( s2 + oa~) J

s4fl(k)

= y o [ ( m s 2 + c ) - k=oOd~ ~ ~f'(k-!-Za2a (s2 +sf2(-k]Oak)2 )] •

(30)

The response of the system, y,., in the time domain can be obtained through inversion. The detail of the mathematical techniques in the inversion process will be discussed in the following section.

Dynamic coupling of a liquid-tank system

269

We next examine a tank which is initially moving with a constant speed, Vo, and is suddenly subjected to a constant force, FB, until it is completely stopped. The motion can be described by the following equation: myc = F n e t - FB.

(31)

The application of the Laplace transform and the substitution of Equation (28) yields the following solution for ~c:

Vo

)7C : ~ -

ms 3 -

~ ~

PB

.

(32)

sSf'(k) - s3f2(k)

k=odd

a2a (s 2 +

o,~,)

Case It--rolling motion of a fluid-tank system By assuming that the ship rolls about a fixed axis through its center of gravity the dynamics of the coupled tank-ship system can be studied with the following equation: I(J + cO + AGMO = rFne t + MI4 + M v + Mext

(33)

where I and A are the transverse mass moment of inertia and the weight of the ship, respectively; c is the damping moment coefficient; r is the vertical distance between the free surface of the undisturbed fluid and the center of gravity of the ship; G M is the metacentric height; Fne t is the horizontal fluid force; M n and M v are the moments about the x-axis of the moving coordinate due to the horizontal and vertical fluid forces, respectively; and Mext is the applied moment due to the wave action. Since the ship is rolling about its center of gravity, the motion of the origin of the moving coordinate is related to the center of gravity of the ship by the following expressions for linear study: ~ = - d0 cos~b = - r 0 ~f~= - d0 sin~b

(34)

where + is the angle measured from the z-axis of the fixed coordinate to the z-axis of the moving coordinate, and d is the distance between the origin of the rotating coordinate to the center of gravity of the ship (see Fig. 1). By giving the ship an initial displacement, 0o, in still water, the transformed equation of Equation (33) can be written as:

O{s3(z- r,- r2) - s3,=,,ddsZ(TI°T" + TgTO+ + S AG-'--M-T3

~ -- T6(D2 +S2( Z 7 T 4 - T6T 8 - Ts(t)2) - s4 T5 Ts] } - k=Zodd (S2+{-02) ]J

=Oo[sZ(1-Tl-T2)-s 2 -

-- T 3 - k=oddZ

+ $2C

s2( TI°T4 + T9Ts)+ TgT6

k=odd

+ sc

(S2+Oj~)

r6,,,~,+s2(r7 r4 - r6r~- T,,o~,)-s4T~ rs] ~

"

J + TII

(35)

27O

A. PuI-CHuNLu] and J. Y. K. Lou

where: 4

3

4

Tt = 2paD2br - 3PDab + jpDba 2 2 7"2 = - paD2rb + ~ paD3b - ~ pDa3b 4

T3 = 4pgDarb - 29gD2ab + 3 pga3b Ta(k)=4pbr( ~2a lsinhe~D-D ) - 4~P~ a5 a ( c o s h a D - 1) a3a _~2

~(k) - -4pbra3a

DasinhaD + ~} coshaD +

-

coshaD + ~

- 4 op3tab (D coshoLD-

r6(k ) -T7(k)-

4pgDbr

2pgD2b

~2a

o~2a

2ga cosh~D

sinhaD

°t21 sinhaD)

(36)

4pgb

+ ot4a

ga

2 Ts(k) - coshoLD T9(k) = - r a tanhaD Tm(k) = - r a 2 Tll = tan0o (2pgDarb - pgDZab + ~ pga3b).

However, if the response due to the wave action is of interest, it can be studied by assuming that the ship is initially at rest in an upright position and is suddenly subjected to an external harmonic wave with wave moment

Me×t =Mocosgt

(37)

where Mo is the amplitude of the wave moment and cr is the frequency of the wave. The substitution of Equation (37) into Equation (33) yields the following equation in the transformed plane

Dynamic coupling of a liqui d-t a nk system

271

~ s2(TloT4+T9Ts)+TgT6 6 s2(1-T,-T2) - s 2 ~ s2+t02 + cs k = odd

+AGM-T3-

~

-,,,~, T6 + s2( TTT,, - T6 Ts- Tso~D- s" Ts T81 ~

j

(38)

k=odd S

= Mo $2+O.2

•

For comparison purposes, the responses of an equivalent non-shifting cargo system are also studied. The equations of motion for the non-shifting cargo system are: (i) for rectilinear motion

msYc + cpc + Kyc = Fext

(39)

(ii) for rolling motion Is0 + cO + (AGM)s0 = Mext

(40)

where ms, Is and (A GM)s are the mass, transverse mass moment of inertia, and the restoring moment coefficient of the rigid-cargo system with equivalent loads. SOLUTION TECHNIQUES AND NUMERICAL RESULTS The study of various types of motion of fluid-tank systems by the Laplace transformation led to the transformed equations whose solution F(s) is of the form

['(s) - H(s) Q(s)

(41)

where H(s) and Q(s) are polynomials in s. The classical method for finding the inverse function involves a contour integration in the complex plane. The value of this contour integral equals the sum of the residues of the function e"F(s). When the singularities are simple poles, the inverse transformation can be accomplished through the use of partial fractions. From Equation (41), it follows that the results of inversion depend on the nature of the roots of the polynomial Q(s). It will, therefore, be of interest to determine how the parameters of the system affect these roots. The stability of motion of the system can be investigated by using the Routh-Hurwitz criterion (Uspensky, 1948). It is also known that if the real parts of the roots of the polynomial Q(s) are negative, the system is dynamically stable. A careful examination of Equations (30), (32), (35), and (38) reveals the complexity of the transformed equations. A closed-form, time domain solution may not be obtained through inversion. Hence, approximate numerical solution will be obtained instead by arbitrary truncating of the series in the transformed equations. Several computer programs have been coded to perform the numerical inversion. Different numbers of terms in the series have been tried to obtain a better numerical solution. The comparison of the numerical results by using different numbers of terms indicate that even though more terms in the series are used, the lower order roots of the polynomial Q(s) remain unchanged. The magnitude of the coefficients in the partial fraction expansion corresponding to the additional roots are very small as compared to the lower order

272

A. PuI-CHuN LuI and J. Y. K. Lou

coefficients. Thus, the contributions of the higher order terms are not significant and the inclusions of these higher order terms can only slightly improve the accuracy of the numerical results at the expense of computer time. A satisfactory solution can be achieved by only including the first five terms in the series in conjunction with partial fraction expansion techniques. Some of the numerical results for the rectilinear and rolling motions of the fluid-tank system are graphically presented in Figs 2-7. These figures also show the comparisons of the responses between the fluid-tank system and the equivalent non-shifting cargo system. Figures 2 and 3 show that responses of the fluid-tank system and equivalent rigidcargo system undergo an oscillatory-type motion. It is noticed from the response curves that the difference in the responses for the two systems depends on the ratio of fundamental natural frequency of the fluid which is given by Lamb, 1945 as

.r,

"IT

to1 = ~/ g 2 a tanh 2a D and the equivalent

"s=

rigid-cargo

(42)

system

•/K-

(43)

m;

3]

I-

2a

I~

"1

~

T

2,0

-

u. 1.0

:

'i

o.o ~'

.~o

:

%%'%\%%

:..

-

" "K/ x /

v

v

-1,0

-2.0

¥

TIME (SEC) (a)

TFlut d / TTank - 0.9

3.0~,

~

FLUID-TANK SYSTEM

2 . 0| ~

.....

RI6/O-CARGO SYSTEM

I-

1.0

L\

o.0 L. ,n o

-1.0

-2.0 10

20

30

TIME (b)

40

(SEC)

TFIU| d / TTzmk - 0.2

FIG. 2. Oscillatory type of motion.

50

Dynamic coupling of a liquid-tank system 51~-r

273

T ~ - 3.96sec

FLUID- TANK SYSTEM

,i

°t t\

--

/A \

T~lp-14.52slc

,.~

RI$1U.CAU$O SYSTEM

/",

A..

N T[HE ISECI

-5 (a)

TFlut d / Tsh4p - O.Z7 Tl,~d=6.50 sec

,.

,~

~.

~.

v,-,

FLUID-TANK SYSTEM

. . . . . . . . . .

...^

.....

"~.

V

~o

3"6-

•

TIME (SEC)

(b) FIG.

TFlu| d / TShtp - 0.94

3. Free roll oscillation of a ship.

where ms is the equivalent total mass. If the ratio is much greater than one, the response of the fluid-tank system behaves like the rigid-cargo system. That is, the motion of the fluid inside the tank has little effect on the system's motion. However, if the ratio is close to one, the effect of fluid motion is very obvious. The fluid-tank system requires more time to damp out the motion as compared to the equivalent rigid-cargo system. A similar result is also obtained for the free roll oscillation of a ship as shown in Fig. 7. Figure 4 presents the motion of a fluid-tank system which is traveling at a constant speed and is suddenly subjected to an applied force until it is completely stopped. It shows very little difference in terms of time and distance required to stop the two systems. Additional computer runs show that the braking performance of the fluid-tank system depends on the amount of fluid inside the tank. A computer run using the fullscale field test data conducted by Culley et al. (1978) shows that the fluid-tank system required more time to reach a complete stop. Compared with the non-shifting cargo system, the fluid-tank system has a poorer braking performance with more fluid. This is consistent with the field test results (Culley et al., 1978). Figures 5 and 6 present the numerical results for the forced roll oscillations of a ship. Figure 5a shows the response of a ship under the excitation of a harmonic wave with a period much greater than the ship's natural period. It is seen that the sloshing liquid has very little effect on the roll motion of the ship, even though the natural period of the fluid is very close to that of the ship. This is because the amount of liquid, for this case, is a small percentage of the total ship displacement. Additional calculations show that if the amount of liquid is increased by increasing the longitudinal tank length, much larger roll amplitudes can be expected. This is shown in Fig. 5b. It should be

274

A. PuI-CHuN Lul and J. Y. K. Lou

_L_,,_ ~ - - F ,

C)

0

\\\",~,\\\NN\\

4.0

2.~ =. 0.{~

<

-2.0

TIME (SEC) -4.0

(a)

5 ~0

~,

TIMEHISTORYOF THE SURFACEELEVATIONAT POINT A

~

FLUID-TANK SYSTEM •

0

'

10

J

20

TIME (SEC) (b) VELOCITYPROFILE TIME (SEC)

: .

.

.

10

.

.

.

.

.

.

(c)

ACCELERATIONPROFILE

Fzo. 4. Fluid-tank system subjected to a constant brake force.

FLUID-TANK SYSTEM .....

TFlut d - 6.50 sec

TWave = 93.17 sec

RIGID-CARGOSYSTEM

Tshtp - 6.95 sec

101 ~

~,,

.

"I ~ -1~

./

.

~,:_iI

.

"~.. ",~

""~ (a)

(b)

.

.

.

.

5o\ /

/

\

'qoo

( WIHmlT OF WATEIt - 2.31"I, OF WEIIIIff OF THE SHIP

v

.-~-20 ,11

.

,. TIME (SEC)

HEITH'rOF HATER . 8.88Y, OF HEIGHT OF THE SHIP

FIG. 5. Forced roll oscillation of a ship, Tnui,i/T.hi p

:

0.94.

D y n a m i c coupling o f a l i q u i d - t a n k s y s t e m

275

FLDIO-TANt( SYSTEM ........ RIGID-CARGO SYSTEM

TFlu4 d =3.96 see A

51

.

.

.

.

.

.

.

t,..,llV,.,o,.ll v

~-5,1

J. ]0

T~ave • 5.00 see

.

z

.

.

TSM p • 14.52 sec

.

.

.

V.

.I.

20

..k

30

50

•

.

.. r v o..v,

.L

.t.

40

z

J-

50

70

.d80

TIHE (SEC)

(a) TWave / T$hlp - 0.33

i

20[

T~,,~ -3.96sec

Tw_.lO.OOsec

T~,-14.52sec

'°r~, / , A , " , A ..,--A .,-,A ~ , A ...., A , ,

-}'-v ,v',v' ,v' v' v v -~o

'..../

-20

:.."

-"10

(b)

~ 20

~ 30

'v"

"...."

~ 40 TIME (SEC)

"...,"

~ 50

..,. 60

.,. 70

THave / TSM p - 0.69

T~-3.96sec

,o[,o

-.J

Tw,,,,-13.00sec

T,~,~ -14.52sec

A / ',,.A ,,f~ /-A/'A/'ol/L.'7 \., 7.\, . l \ , . , l \ , . ¢.\

-10

•

~

t

~

:

"

•

~

f" V V U U'

.20 L

x...

\

.J.

10

(c)

;

;

"

~

"..y

~

20

"

i

-/

~

30

,

"

.,.

"

:

\.]

40 TIME (SEC)

.,"

.~.

50

:

~ \

l

t

:

' \

"-...../

-

:

/~

-.../

~

60

70

THovo / TSh4p " 0.9

FIG. 6. F o r c e d roll oscillation o f a ship.

noted that for the case shown in Fig. 5b, the amount of liquid is about four times that of Fig. 5a but is still very small compared to the ship's displacement. Figure 6 demonstrates the influence of forcing frequency on sloshing effects. All the physical parameters, such as ship properties, tank geometry, natural periods, etc. are fixed for the three cases. Only the forcing frequency changes. It is seen that the effects of liquid sloshing become more pronounced as the wave frequency approaches more closely the natural frequency of the fluid. The time history of the surface elevation of the uncoupled force roll oscillation is obtained by using the linear analysis. The fluid is forced to oscillate about an axis on the center bottom of the tank with the forcing frequency equal to the second natural frequency of the fluid. The linear analysis result is shown in Fig. 7 along with the experimental results conducted by Lou et al. (1980) and the numerical results of Bridges (1981) by using a finite difference technique (MAC method). The result from the linear analysis is in very good agreement with the other two methods except that the magnitude

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is slightly larger. It should be noted, however, that the experimental result was recorded at a slightly different scale both for the magnitude and for the time. CONCLUSION The analytical m e t h o d developed in this study provides a simple and quick technique to examine the dynamic coupling effects of the fluid-tank system. T h e numerical solutions show a distinctive difference of responses between the fluid-tank system and the non-shifting rigid-cargo system when the fluid natural frequency is close to the natural frequency of the tank. It also indicates that the responses of the fluid-tank system d e p e n d on the a m o u n t of fluid inside the tank and the initial stability of the system. H o w e v e r , this study fails to provide a p r o p e r solution when the amplitude of m o t i o n is large and the water depth is too shallow due to the limitation of linear analysis. If the response of the fluid-tank system with large amplitude m o t i o n or shallow water depth is of interest, it can be studied by including the nonlinear terms in the governing equations. In this case, the time domain solution has to be formulated since the Laplace transformation technique is not valid for nonlinear studies. REFERENCES ABRAMSON,H.N. 1966. The dynamic behavior of liquids in moving containers with application to space vehicle technology. NASA-SP-106, National Aeronautics and Space Administration, Washington, D.C. BRIDGES, T. 1981. Private communication. CHU, W.H., DALZELL,J.F. and MODISETTE,J.E. 1968. Theoretical and experimental study of ship-roll stabilization tanks. J. Ship. Res. 12, 168-181. Cox, P.A., BOWLES,E.B. and BASS, R.L. 1979. Evaluation of liquid dynamic loads in slack LNG cargo tanks. Southwest Research Institute Report No. SR-1251, San Antonio, Texas. CULLEY, C., ANDERSON,R.L. and WESSON,L.E. 1978. Effect of cargo shifting on vehicle handling. DOT Report No. FHWA-RD-78-76, Dynamic Science Inc., Phoenix, Arizona, U.S.A. HAMLIN, N.A., Lou, Y.K., MACLEAN,W.M., SEIBOLD, F. and CHANDRAS,L.M. 1986. Liquid sloshing in slack ship tanks--theory, observations and experiments. Trans. Soc. nay. Archit. mar. Engrs, 94, 159-195. LAMB, H. 1945. Hydrodynamics (6th Edition). Dover Publications, New York.

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Lou, Y.K., Su, T.C. and FLIPSE, J.E. 1980. A nonlinear analysis of liquid sloshing in rigid containers. Texas A & M University, Research Foundation Report, Contract No. DOT-RC-92037. THOMSON, W.T. 1950. Laplace Transformation Theory and Engineering Applications. Prentice-Hall, New York. USPENSKV, J.V. 1948. Theory of Equations. McGraw-Hill, New York. VISINEAU, G. 1979. Relative roll motion of a ship in beam seas. Presented to Northern California Section of SNAME. SNAME, Northern California Section.